The difference between reversible
and irreversible processes is brought out through examination of the
isothermal expansion of an ideal gas. The question to be asked is
what is the difference between the ``free expansion'' of a gas and
the isothermal expansion against a piston? To answer this, we
address the steps that we would have to take to reverse, in other
words, to undo the process.

By free expansion, we mean the unrestrained expansion of a gas into
a volume as shown in Figure 4.3. (The
restrained expansion is shown in
Figure 4.4.) Initially all the gas
is in the volume designated as
with the rest of the insulated
enclosure a vacuum. The total volume (
plus the evacuated
volume) is
. At a given time a hole is opened in the partition
and the gas rushes through to fill the rest of the enclosure.

Figure 4.3:
Free expansion

Figure 4.4:
Expansion against a piston

During the expansion there is no work exchanged with the
surroundings because there is no motion of the
boundaries4.1. The enclosure is insulated
so there is no heat exchange. The first law tells us therefore that
the internal energy is constant (
). For an ideal gas,
the internal energy is a function of temperature only so that the
temperature of the gas before the free expansion and after the
expansion has been completed is the same. Characterizing the before
and after states:

Before: State 1,
,

After: State 2,
,
.

, so there is no change in the surroundings.

To restore the original state, i.e., to go back to the original
volume at the same temperature (
at constant
)
we can compress the gas isothermally (using work from an external
agency). We can do this in a quasi-equilibrium manner, with
, as in
Figure 4.5. If so the work that we need
to do is
. We have evaluated the work in a
reversible isothermal expansion (Eq. 3.1), and
we can apply the arguments to the case of a reversible isothermal
compression. The work done on the system to go from state
``2'' to state ``1'' is

Figure 4.5:
Returning the free expansion to its initial condition

From the first law, this amount of heat must also be rejected from
the gas to the surroundings if the temperature of the gas is to
remain constant. A schematic of the compression process, in terms of
heat and work exchanged is shown in
Figure 4.6.

Figure 4.6:
Work and heat exchange in the
reversible isothermal compression process

At the end of the combined process (free expansion plus
reversible compression):

The system has been returned to its initial state (no change in
system state).

The sum of all of these events
is that we have converted an amount of work,
, into an amount of
heat,
, with
and
numerically equal in Joules.

The net effect is the same as if we let a weight fall and pull a
block along a rough surface, as in Figure 4.7. There is
100% conversion of work into heat.

Figure 4.7:
100% conversion of work into heat

The results of the free expansion can be contrasted against a
process of isothermal expansion against a pressure
which is
slightly different than that of the system, as shown in
Figure 4.8.

Figure 4.8:
Work and heat transfer in
reversible isothermal expansion

During the expansion, work is done on the surroundings of magnitude
, where
can be taken as the system pressure. As
evaluated in Eq. (3.1), the magnitude of the
work done by the system is
. At the end of the isothermal
expansion, therefore:

The surroundings have received work
.

The surroundings have given up heat,
, numerically equal to
.

We now wish to restore the system to its initial state, just as we
did in the free expansion. To do this we need to do work on the
system and extract heat from the system, just as in the free
expansion. In fact, because we are doing a transition between the
same states along the same path, the work and heat exchange are the
same as those for the compression process examined just above.

The overall result when we have restored the system to the initial
state, however, is quite different for the reversible expansion than
for the free expansion. For the reversible expansion, the work we
need to do on the system to compress it has the same magnitude as
the work we received during the expansion process. Indeed, we could
raise a weight during the expansion and then allow it to be lowered
during the compression process. Similarly the heat put into the
system by us (the surroundings) during the expansion process has the
same magnitude as the heat received by us during the compression
process. The result is that when the system has been restored
back to its initial state, so have the surroundings. There is no
trace of the overall process on either the system or the
surroundings. That is another meaning of the word ``reversible.''

Muddy Points

With the isothermal reversible expansion is
constant? If so, how can we have
?
(MP 4.2)